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GaussInt

 GIfactor
 Gaussian integer factorization

 Calling Sequence GIfactor(c)

Parameters

 c - Gaussian integer, a list or a set of Gaussian integers

Description

 • The GIfactor function returns the Gaussian integer factorization of c.
 • The answer is in the form: $u{\left({f}_{1}\right)}^{{e}_{1}}\cdot \dots \cdot {\left({f}_{n}\right)}^{{e}_{n}}$ such that $c=u\cdot {f}_{1}^{{e}_{1}}\cdot \dots \cdot {f}_{n}^{{e}_{n}}$ where $u$ is one of four units, ${f}_{1},...,{f}_{n}$ are the distinct primary prime factors of c, and ${e}_{1},...,{e}_{n}$ are their multiplicities.
 • The number $x+Iy$ of the four associated primes $a+Ib$, $-a-Ib$, $-b+Ia$, $b-Ia$ is singled out as primary in which we have simultaneously either ($x=1\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}4\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}y=0\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}4$) or ($x=-1\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}4\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}y=2\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}4$).
 • The expand function may be applied to cause the factors to be multiplied together again.

Examples

 > $\mathrm{with}\left(\mathrm{GaussInt}\right):$
 > $\mathrm{GIfactor}\left(-72-100I\right)$
 $\left({I}\right){}{\left({1}{+}{I}\right)}^{{4}}{}\left({-3}{-}{2}{}{I}\right){}\left({-3}{+}{8}{}{I}\right)$ (1)
 > $\mathrm{expand}\left(\right)$
 ${-72}{-}{100}{}{I}$ (2)
 > $\mathrm{GIfactor}\left(12101-15295I\right)$
 $\left({-1}\right){}\left({1}{+}{I}\right){}\left({1}{+}{4}{}{I}\right){}\left({5}{+}{8}{}{I}\right){}\left({9}{+}{16}{}{I}\right){}\left({-7}{-}{18}{}{I}\right)$ (3)
 > $\mathrm{expand}\left(\right)$
 ${12101}{-}{15295}{}{I}$ (4)

 See Also